The kernel density estimates are probability density functions (PDFs) and belong to the Banach manifold of all PDFs. We focus on PDFs on the domain \([0,1]\) , however the following description can be applied to other domains with simple adjustments. Let \(\mbox{$\mathcal {F}$}= \big \lbrace f:[0,1] \rightarrow \mathbb {R}_{> 0} \big | \int _0^1 f(x) dx = 1 \big \rbrace \) . We use the Fisher-Rao (F-R) Riemannian metric [1]}, [2]}, [3]} on \(\mbox{$\mathcal {F}$}\)  to make it a Riemannian manifold and facilitate computation. This metric has nice statistical properties such as invariance to bijective and smooth transformations of the PDF domain [4]}; it is also closely related to the Fisher information matrix. However, the metric is difficult to employ for computation so we skip the specific formula here for brevity. To build some basic statistical tools, e.g., the distance between a pair of densities, Karcher mean for a sample of densities, and PCA on the space of densities, we work on a transformed space of PDFs using a square-root transformation [5]}. Consequently, the space of PDFs becomes the positive orthant of the unit sphere in \(\mathbb {L}^2\) , whose geometry is well-known, and the F-R metric flattens to the standard \(\mathbb {L}^2\)  metric.
